Discrete Optimization 2 (2005) 113–122www.elsevier.com/locate/disopt
A semideﬁnite optimization approach for the singlerow layoutproblem with unequal dimensions
Miguel F.Anjos
a
,
∗
,
1
,Andrew Kennings
b
,
2
,Anthony Vannelli
b
,
3
a
Operational Research Group, School of Mathematics, University of Southampton, Highﬁeld, Southampton SO17 1BJ, UK
b
Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1
Received 2April 2004; received in revised f orm 29 March 2005; accepted 31 March 2005
Abstract
Thefacilitylayoutproblemisconcernedwiththearrangementofagivennumberofrectangularfacilitiessoastominimizethetotalcostassociatedwiththe(knownorprojected)interactionsbetweenthem.Weconsidertheonedimensionalspaceallocationproblem (ODSAP), also known as the singlerow facility layout problem, which consists in ﬁnding an optimal linear placementof facilities with varying dimensions on a straight line. We construct a semideﬁnite programming (SDP) relaxation providing alower bound on the optimal value of the ODSAP.To the best of our knowledge, this is the ﬁrst nontrivial global lower bound fortheODSAPinthepublishedliterature.ThisSDPapproachimplicitlytakesintoaccountthenaturalsymmetryoftheproblemand,unlike other algorithms in the literature, does not require the use of any explicit symmetrybreaking constraints. Furthermore,the structure of the SDP relaxation suggests a simple heuristic procedure which extracts a feasible solution to the ODSAP fromthe optimal matrix solution to the SDP relaxation. Computational results show that this heuristic yields a solution which isconsistently within a few percentage points of the global optimal solution.© 2005 Elsevier B.V.All rights reserved.
Keywords:
Facilities planning and design; Space allocation; Combinatorial optimization; Semideﬁnite optimization; Global optimization
1. Introduction
The facility layout problem is concerned with the arrangement of a given number of rectangular facilities so as to minimizethe total cost associated with the (known or projected) interactions between them.Versions of the facility layout problem occurin many environments, such as hospital layout and service center layout. This is a hard problem in general; most versions of thisproblem in the research literature are known to be NPhard. A thorough survey of the facility layout problem is given in [24],where the research papers on facility layout are divided into three broad areas.The ﬁrst is concerned with algorithms for tackling
∗
Corresponding author. Present address: Department of Management Sciences, University of Waterloo, Waterloo, Ont., Canada N2L 3G1.
Email addresses:
anjos@stanfordalumni.org (M.F.Anjos), akenning@cheetah.vlsi.uwaterloo.ca (A. Kennings),
vannelli@cheetah.vlsi.uwaterloo.ca (A.Vannelli).
1
ResearchofM.F.AnjoswaspartiallysupportedbyGrantNAL/00636/GfromtheNufﬁeldFoundationandGrantA2002/3fromtheUniversityof Southampton.
2
Research ofA. Kennings was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada DiscoveryGrant 20376303.
3
Research ofA.Vannelli was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada OperatingGrant 15296 and a Bell University Laboratories Research Grant.15725286/$see front matter © 2005 Elsevier B.V.All rights reserved.doi:10.1016/j.disopt.2005.03.001
114
M.F. Anjos et al./Discrete Optimization 2 (2005) 113–122
the general layout problem as deﬁned above.The second area is concerned with extensions of the problem in order to account foradditional issues which arise in applications, such as designing dynamic layouts by taking time dependency issues into account;designing layouts under uncertainty conditions; and achieving layouts which optimize two or more objectives simultaneously.The third area is concerned with specially structured instances of the problem. One such special case that has been extensivelystudied occurs when all the facilities have equal dimensions and the possible locations for the facilities are given a priori; this isthe quadratic assignment problem (QAP) formulated by Koopmans and Beckman [19]. The QAP assigns every facility to one
location and at most one facility to each location, and the cost of placing a facility at a particular location is dependent on thelocation of the interacting departments. Since the possible locations are ﬁxed, the problem reduces to optimizing a quadraticobjective over all possible assignments of facilities to locations. The QAP is NPhard, and is in general a hard problem to solve.Indeed, the wellknown Nugent instances of this problem with up to 30 departments were solved to proven optimality in [3]using vast amounts of computational power and important improvements in mathematical programming algorithms.In this paper, we consider a related special case of facility layout, namely the placement of facilities of given, and possiblydifferent, dimensions on a straight line. This problem is known in the literature both as the linear singlerow facility layoutproblem, see for example [14], and as the onedimensional space allocation problem (ODSAP), see [29]. We shall refer to it as
theODSAP.AninstanceoftheODSAPconsistsof
n
onedimensionalfacilities
{
r
1
,...,r
n
}
forwhichwearegiventheirpositivelengths
ℓ
1
,...,ℓ
n
as well as their pairwise connectivities,
c
ij
. We are interested in ﬁnding an arrangement of the facilities nexttoeachotheralong aline so astominimize the total weightedsum of thecentertocenter distancesbetween all pairs of facilities.Therefore, like the aforementioned QAP, the ODSAP is an optimization problem over all possible permutations of the givenfacilities. In the special case where all the facilities have the same length, the ODSAP becomes a special case of the QAP [1,22].Several applications of the ODSAP have been identiﬁed in the literature. These include the arrangement of books on a shelf,the layout of warehouses, and the layout of machines on a factory ﬂoor [14,26]. Furthermore, the ODSAP is closely relatedto the linear ordering problem, which also has a number of practical applications. A summary of these can be found in [8],along with references to the relevant literature. It is worth pointing out that the model we introduce in Section 2 captures thesame combinatorial structure as the linear ordering polytope studied in [9,27], except that our model is based on semideﬁniteprogramming (SDP) rather than linear programming.Let
=
(
1
,...,
n
)
denote a permutation of the indices
[
n
] := {
1
,
2
,...,n
}
of the facilities, so that the leftmost facility is
r
1
, the facility to the right of it is
r
2
, and so on, with
r
n
being the last facility in the arrangement. Given a permutation
andtwo distinct facilities
r
i
and
r
j
, the centertocenter distance between
r
i
and
r
j
with respect to this permutation is
12
ℓ
i
+
D
(i,j)
+
12
ℓ
j
,where
D
(i,j)
denotes the sum of the lengths of the facilities between
r
i
and
r
j
in the linear arrangement deﬁned by
. Tosolve the ODSAP, we seek a permutation of the facilities which minimizes the weighted sum of the distances between all pairsof facilities. In mathematical terms, we wish tomin
∈
i<j
c
ij
12
ℓ
i
+
D
(i,j)
+
12
ℓ
j
,where
denotes the set of all permutations
of
[
n
]
.The ODSAP was ﬁrst studied by Simmons [29] who observed that it is possible to simplify the objective by eliminating thehalffacility lengths. Indeed, we can rewrite the objective function asmin
∈
i<j
c
ij
D
(i,j)
+
i<j
12
c
ij
(ℓ
i
+
ℓ
j
)
,where the second summation is a constant independent of
. We will thus focus our attention on optimizing
i<j
c
ij
D
(i,j)
over all permutations
.Another observation concerns the symmetry of the arrangements. It is clear that
D
(i,j)
=
D
′
(i,j)
,where
′
denotes the permutation symmetric to
, deﬁned by
′
(i)
=
(n
+
1
−
i)
for
i
=
1
,...,n
.It follows that for the ODSAP, we can exchange the left and right ends of the layout and obtain the same objective value. Hence,it possible to simplify the problem by considering only the permutations for which, say,
r
1
is on the left half of the arrangement.Thistypeofsymmetrybreakingstrategyisimportantforreducingthecomputationalrequirementsofmostalgorithms,including
M.F. Anjos et al./Discrete Optimization 2 (2005) 113–122
115
those based on linear programming or dynamic programming. However, our proposed approach will implicitly consider thesesymmetries and does not require the use of any explicit symmetrybreaking constraints.Several algorithms have been proposed for solving the ODSAP. Simmons [29] proposed a branchandbound algorithm; Loveand Wong [23] considered a mixed integer linear programming model; and Picard and Queyranne [26] developed a dynamic
programming algorithm, extending an algorithm of Karp and Held [16] for the special case where all the facilities have equallengths. All these algorithms are guaranteed to ﬁnd the global optimal solution, but they have very high computational timesand memory requirements, and are unlikely to be effective for problems with 20 or more facilities. Research has also been doneon heuristic algorithms, which are more efﬁcient but provide no guarantee of global optimality. We point out the recent work of Heragu and Kusiak [15], who tackled the ODSAP using nonlinear optimization methods; Romero and SánchezFlores [28] and
Heragu and Alfa [13], who developed simulated annealing algorithms; and Kumar et al. [20], who proposed a greedy heuristic
algorithm.In this paper, we propose the application of SDP techniques to the ODSAP. SDP refers to the class of optimization problemswhere a linear function of a matrix variable
X
is maximized (or minimized) subject to linear constraints on the elements of
X
andthe additional constraint that
X
be positive semideﬁnite. This includes linear programming problems as a special case, namelywhen all the matrices involved are diagonal.A variety of polynomialtime interiorpoint algorithms for solving SDPs have beenproposed in the literature, and several excellent solvers for SDP are now available. We refer the reader to the Handbook [32] fora thorough coverage of the theory and algorithms in this area, as well as several application areas where SDP researchers havemade signiﬁcant contributions. In particular, SDP has been very successfully applied to problems which, like the ODSAP, havea strong combinatorial ﬂavor. Recent survey papers on the application of SDP to combinatorial optimization include [2,21].We construct an SDP relaxation providing a lower bound on the optimal value of the problem. To the best of our knowledge,this is the ﬁrst nontrivial global lower bound for the ODSAP in the published literature. In particular, our relaxation implicitlytakes into account the natural symmetry of the problem and, unlike other algorithms in the literature, does not require the useof any explicit symmetrybreaking constraints. Furthermore, the structure of the SDP relaxation suggests a simple heuristicprocedure which extracts a feasible solution to the ODSAP from the optimal matrix solution to the SDP relaxation. Therefore,our approach yields both a feasible solution to the given ODSAP instance as well as a guarantee of how far it is from globaloptimality. When applied to problems previously considered in the literature, our lower bounds also provide a measure of thedistance from optimality of the best layouts obtained using some of the aforementioned heuristics.This paper is structured as follows. In Section 2, we present a formulation of the ODSAP over the space of real symmetricmatrices,proveitscorrectness,andtherebyderivetheSDPrelaxationfortheODSAP.InSection3,wepresentthesimpleheuristicthat extracts a speciﬁc layout from the optimal solution to the SDP. In Section 4, we apply our algorithm to some instances of the ODSAP (with up to 30 facilities) which have been previously considered in the literature. This provides a comparison of the SDP approach with previous approaches. We also show how the global lower bound provided by the SDP relaxation canbe improved by using a simple branching step and solving two or more additional SDPs. Finally, we apply the SDP approachto randomly generated instances with up to 80 facilities and obtain layouts that are consistently a few percentage points fromglobal optimality.
2. SDP relaxation of the ODSAP
As mentioned above, the ODSAP is essentially a problem over all possible permutations of the
n
facilities. For each pair
i,j
of facilities with
i <j
, deﬁne the binary
±
1 variable
R
ij
:=
1 if facility
i
is to the right of facility
j,
−
1 if facility
i
is to the left of facility
j.
It is clear that exactly one of these two possibilities must hold for every feasible allocation. The order of the subscripts matters,and it is obvious that
R
ij
=−
R
ji
. Since two symmetric representations exist for each arrangement, we may require that
R
ij
=
1for a speciﬁed pair
(i,j)
to eliminate this symmetry. However, unlike most other formulations for layout problems, the modelwe propose implicitly takes into account the existence of symmetries without requiring the addition of such symmetrybreakingconstraints.To accurately formulate the problem, it is not enough to require that all the
R
ij
equal
±
1. We must also ensure that theyrepresent a valid arrangement of the
n
facilities. In particular, we require thatif
R
ij
=
R
jk
,
then
R
ik
=
R
ij
,which we can formulate as
(R
ij
+
R
jk
)(R
ik
−
R
ij
)
=
0,
116
M.F. Anjos et al./Discrete Optimization 2 (2005) 113–122
which expanded yields the quadratic constraint
R
ki
R
ij
−
R
ij
R
kj
−
R
ki
R
kj
= −
1.In principle, we have three such constraints for each triple
(i,j,k)
, but it is straightforward to check that they are all equivalentto this single quadratic equation with
i <j <k
. Therefore, we can reorder indices and express these necessary constraints as
R
ij
R
jk
−
R
ij
R
ik
−
R
ik
R
jk
= −
1 for all triples
i <j <k
. (1)We now prove that these
n
3
constraints on the
R
ij
variables sufﬁce to represent all possible permutations of the
n
facilities.Let
∈ {±
1
}
(
n
2
)
denote a particular assignment of values to the
R
ij
variables such that the conditions (1) are satisﬁed, and let
R
denote the set of all such
R
:= {
∈ {±
1
}
(
n
2
)

R
ij
R
jk
−
R
ij
R
ik
−
R
ik
R
jk
= −
1 for all triples
i <j <k
}
.We show how each permutation
∈
corresponds to a unique
∈
R
, and vice versa.I.
From
to
: Let
=
(i
1
,i
2
,...,i
n
)
be any permutation of the integers
[
n
]
. Set
R
i
p
,i
q
= −
1 for all
p <q
.(Note that
i
p
>i
q
may hold even if
p <q
, and if that is the case, then
R
i
q
,i
p
=
1.) Then
i
1
is the leftmost facility, with
i
2
onits right, and so on, up to
i
n
being the rightmost facility. This is the desired representation of
.II.
From
to
: Given
∈
R
, consider
P
k
=
j
=
k
R
kj
=
j<k
−
R
jk
+
j>k
R
kj
for
k
=
1
,
2
,...,n
. (2)Clearly all the
P
k
values are integer and belong to the set
P
:= {−
(n
−
1
),
−
(n
−
3
),...,n
−
3
,n
−
1
}
which has exactly
n
elements.A straightforward mapping of the elements of
P
onto
[
n
]
is given by
p
k
=
P
k
+
n
+
12 .We prove that no two
P
k
values are equal. This implies that every element of
P
is represented by exactly one
P
k
, and hencethat
(p
1
,p
2
,...,p
n
)
is a permutation of
[
n
]
representing
.
Theorem 1.
If
∈
R
then the values
P
k
deﬁned in
(2)
are all distinct
.
Proof.
The proof is by contradiction. Suppose that
P
k
1
=
P
k
2
for
k
1
=
k
2
. Without loss of generality, we can assume
k
1
<k
2
.Then by (2), we have
R
k
1
,k
2
+
k<k
1
−
R
k,k
1
+
k
1
<k<k
2
R
k
1
,k
+
k
2
<k
R
k
1
,k
= −
R
k
1
,k
2
+
k<k
1
−
R
k,k
2
+
k
1
<k<k
2
−
R
k,k
2
+
k
2
<k
R
k
2
,k
.Multiplying on both sides by
R
k
1
,k
2
, we obtain1
+
k<k
1
−
R
k,k
1
R
k
1
,k
2
+
k
1
<k<k
2
R
k
1
,k
R
k
1
,k
2
+
k
2
<k
R
k
1
,k
R
k
1
,k
2
= −
1
+
k<k
1
−
R
k,k
2
R
k
1
,k
2
+
k
1
<k<k
2
−
R
k,k
2
R
k
1
,k
2
+
k
2
<k
R
k
2
,k
R
k
1
,k
2
since
R
2
k
1
,k
2
=
1. Therefore,
k<k
1
(
−
R
k,k
2
R
k
1
,k
2
+
R
k,k
1
R
k
1
,k
2
)
+
k
1
<k<k
2
(
−
R
k,k
2
R
k
1
,k
2
−
R
k
1
,k
R
k
1
,k
2
)
+
k
2
<k
(R
k
2
,k
R
k
1
,k
2
−
R
k
1
,k
R
k
1
,k
2
)
=
2.
M.F. Anjos et al./Discrete Optimization 2 (2005) 113–122
117
Now, using the quadratic constraints from (1), we have that
−
R
k,k
2
R
k
1
,k
2
+
R
k,k
1
R
k
1
,k
2
= −
1
+
R
k,k
1
R
k,k
2
,
−
R
k,k
2
R
k
1
,k
2
−
R
k
1
,k
R
k
1
,k
2
= −
1
−
R
k
1
,k
R
k,k
2
,
R
k
2
,k
R
k
1
,k
2
−
R
k
1
,k
R
k
1
,k
2
= −
1
+
R
k
1
,k
R
k
2
,k
and thus
k<k
1
(
−
1
+
R
k,k
1
R
k,k
2
)
+
k
1
<k<k
2
(
−
1
−
R
k
1
,k
R
k,k
2
)
+
k
2
<k
(
−
1
+
R
k
1
,k
R
k
2
,k
)
=
2,which is equivalent to
k<k
1
R
k,k
1
R
k,k
2
−
k
1
<k<k
2
R
k
1
,k
R
k,k
2
+
k
2
<k
R
k
1
,k
R
k
2
,k
=
n
.But since the lefthand side is bounded above by
n
−
2, we have a contradiction.
It remains to express the objective function of the ODSAP in terms of the variables
R
ij
. It sufﬁces to observe that the sum of the lengths of the facilities between
i
and
j
can be expressed as
k
=
i,j
ℓ
k
1
−
R
ki
R
kj
2
.The justiﬁcation for this fact is the observation that
k
is between
i
and
j
if and only if
R
ki
= −
R
kj
holds, or equivalently,
R
ki
R
kj
= −
1. The objective is thus to minimize
i<j
c
ij
ℓ
i
+
ℓ
j
2
+
k
=
i,j
ℓ
k
1
−
R
ki
R
kj
2
=
i<j
c
ij
n
k
=
1
ℓ
k
2
−
k
=
i,j
ℓ
k
R
ki
R
kj
2
=
i<j
c
ij
2
n
k
=
1
ℓ
k
−
i<j
c
ij
2
k<i
ℓ
k
R
ki
R
kj
−
i<k<j
ℓ
k
R
ik
R
kj
+
k>j
ℓ
k
R
ik
R
jk
and hence our formulation of the ODSAP ismin
K
−
i<j
c
ij
2
k<i
ℓ
k
R
ki
R
kj
−
i<k<j
ℓ
k
R
ik
R
kj
+
k>j
ℓ
k
R
ik
R
jk
s
.
t.
R
ij
R
jk
−
R
ij
R
ik
−
R
ik
R
jk
= −
1 for all triples
i <j <k
,
R
2
ij
=
1 for all
i <j
,where
K
=
i<j
c
ij
2
n
k
=
1
ℓ
k
.We point out that if every
R
ij
variable is replaced by its negative, then there is no change whatsoever to the formulation. Thisis how our formulation, and the subsequent SDP relaxation, implicitly take into account the natural symmetry of the ODSAP.
2.1. Formulation in matrix space and SDP relaxation
Since our objective is to apply SDP to the ODSAP, the next step is to formulate the ODSAP in the space of real symmetricmatrices. Let
P
denote the set of all pairs
(i,j)
such that
i <j
, thus the cardinality of
P
is
n
2
. Deﬁne the vector
v
of length
n
2
v
:=
(R
p
1
,...,R
p
( n
2
)
)
T